# Prokhorov-like conditions for weak compactness of sets of bounded Radon measures on different topological spaces

## TL;DR

This paper extends weak compactness criteria for bounded Radon measures to general topological spaces, using a new function space to replace the classical approach that relies on continuous functions.

## Contribution

It introduces a Prokhorov-like criterion for weak compactness of Radon measures on arbitrary topological spaces, utilizing metasemicontinuous functions instead of continuous functions.

## Key findings

- Established a weak compactness criterion similar to Prokhorov's for general topological spaces.
- Defined a new function space $S(T,\mathcal{G})$ of metasemicontinuous functions.
- Provided conditions under which sets of Radon measures are weakly compact in this setting.

## Abstract

The paper presents some weak compactness criterion for a subset $M$ of the set $\mathfrak{RM}_b(T,\mathcal{G})$ of all positive bounded Radon measures on a Hausdorff topological space $(T,\mathcal{G})$ similar to the Prokhorov criterion for a complete separable metric space. Since for a general topological space the classical space $C_b(T,\mathcal{G})$ of all bounded continuous functions on $T$ can be trivial and so does not separate points and closed sets, instead of $C_b(T,\mathcal{G})$-weak compactness we consider $S(T,\mathcal{G})$-weak compactness with respect to the new uniformly closed linear space $S(T,\mathcal{G})$ of all (symmetrizable) metasemicontinuous functions.